15

The difference of two natural numbers is 5 and the difference of their reciprocals is 5/14. Find the numbers.

Let the required natural numbers x and (x + 5)

x < x + 5

Thus

According to given condition,

taking LCM

cross multiplying

Using the splitting middle term – the middle term of the general equation is divided in two such values that:

Product = a.c

For the given equation a = 1 b = 5 c = – 14

= 1. – 14 = – 14

And either of their sum or difference = b

= 5

Thus the two terms are 7 and – 2

Difference = 7 – 2 = 5

Product = 7. – 2 = – 14

x (x + 7) – 2(x + 7) = 0

(x – 2) (x + 7) = 0

(x – 2) = 0 or (x + 7) = 0

x = 2 or x = – 7

x = 2 (x < x + 3)

x + 5 = 2 + 5 = 7

Hence required natural numbers are 2 and 7.

15

16

The sum of the squares of two consecutive multiples of 7 is 1225. Find the multiples.

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